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%I #10 May 31 2022 02:39:32
%S 1,1,4,16,83,526,3826,31338,285556,2857831,31083421,364523891,
%T 4579906098,61313286380,870531542926,13055593578453,206097824225131,
%U 3414146518958089,59189048364709453,1071264611091540458,20197719805598878119,395917304689782855768
%N G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)^3) / (1 - x)^3.
%H Seiichi Manyama, <a href="/A351816/b351816.txt">Table of n, a(n) for n = 0..486</a>
%F a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n+2*k+1,n-k-1) * a(k).
%t nmax = 21; A[_] = 0; Do[A[x_] = 1 + x A[x/(1 - x)^3]/(1 - x)^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 2 k + 1, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
%Y Cf. A000110, A045501, A125274, A351813, A351817, A351818.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Feb 20 2022